Vector Theory E

8/12/2019 Vector Theory E

1/26

FIZD_

"filcsdgufirpdyscos.cl" 9

^hoter^ho ters ilj tdhcr rhprhshlti tc el =

^hoter quiltctchs irh sphocnchj by jhnclcth fiklctujh ilj jhnclcth jcrhotcel. I vhoter cs khlhriaay

rhprhshlthj by i jcrhothj aclh shkfhlt, siy IB. I cs oiaahj tdh clctcia peclt ilj B cs oiaahj tdh

thrfclia peclt. Zdh fiklctujh en vhoter IB cs hxprhsshj byIB.

]hre vho te r=I vhoter en zhre fik lctujh c.h. wdcod dis tdh sifh clct cia ilj thrf clia peclt, cs oiaahj izhre vhoter.

Ct cs jhlethj by E . Zdh jcrhotcel en zhre vhoter cs cljhthrfclith.

Xlct vho te r=I vhoter en ulct fiklctujh cl tdh jcrhotcel en i vhoter i

cs oiaahj ulct vhoter iaelk i

ilj cs jhlethj by

i , syfbeacoiaay|i|

ii

.

Hxifpah # 9 = Nclj ulct vhoter en g?m2c

_eautcel = i

6 g

?m

2c

cn i

6 cix + miy + giz

tdhl |i|

6 2

z2

y2

x iii

|i|

6 9:

i 6 |i|

i

69:

9c

9:

2m +

9:

?g

Hquia vhoters =Zwe vhoters irh sicj te bh hquia cn tdhy divh tdh sifh fiklctujh, jcrhotcel ilj rhprhshlt tdh sifh

pdyscoia quiltcty.

Oeaa clhir vhoters=Zwe vhoters irh sicj te bh oeaaclhir cn tdhcr jcrhothj aclh shkfhlts irh piriaaha crrhsphotcvh en tdhcr

jcrhotcels. Oeaaclhir vhoters irh iase oiaahjpiriaaha vhoters. Cn tdhy divh tdh sifh jcrhotcel tdhy irh

lifhj isacgh vhotersetdhrwcshulacgh vhoters.

_yfbeacoiaay, twe lel-zhre vhotersiilj

b irh oeaaclhir cn ilj elay cn, bi

, wdhrh R

bi

g

im

ic

i ?29 6 g

bm

bc

b ?29 i96b9, i 26b2, i ?6b?


2/26

FIZD_


9

9

b

i6

2

2

b

i6

?

?

b

i( 6)

^hoters i

6 i9c + mi2 + gi? ilj b

6 b

9c + mb2 + g

b? irh oeaaclhir cn9

9

b

i6

2

2

b

i6

?

?

b

i

Hxifpah # 2 = Nclj viauhs en x & y ner wdcod tdh vhotersi

6 (x + 2)c (x y) m + g

b

6 (x 9)c + (2x + y)m + 2 g irh piriaaha.

_eautcel = i

ilj b

irh piriaaha cn9x

2x

6yx2

xy

62

9

x 6 8, y 6 20

Oepailir vhoters=I kcvhl lufbhr en vhoters irh oiaahj oepa ilir cn tdhcr aclh shkf hlts irh iaa piriaaha te tdh sifh pailh.

Leth tdit twe vhoters irh iawiys oepailir.

Fuatcpacoitcel en i vhoter by i soiair =Cn i

cs i vhoter ilj f cs i soiair, tdhl f i

cs i vhoter piriaaha te i

wdesh fiklctujh csftcfhs tdit

en i

. Zdcs fuatcpacoitcel cs oiaahj soiair fuatcpacoitcel. Cn i

ilj b

irh vhoters ilj f, l irh soiairs,

tdhl =

iff)i()i(f

, i)fl()if(l)il(f

ilifi)lf(

, bfif)bi(f

_han Priotcoh Prebahfs =

(9) Kcvhl i rhkuair dhxikel IBOJHN wctd ohltrh E, sdew tdit

(c) EB EI 6 EO EJ (cc) EJ + EI 6 2 EB + EN

(ccc) IJ + HB + PO 6 : IB

(2) Zdh vhoter gmc bcshots tdh ilkah bhtwhhl tdh vhoters o

ilj m:c? . Jhthrfclh tdh ulct

vhoter iaelk o

.

(?) Zdh suf en tdh twe ulct vhoters cs i ulct vhoter. _dew tdit tdh fiklctujh en tdh tdhcr jcnnhrhloh

cs ? .

Ilswhrs = (2) g98

9:m

98

2c

?

9

Ijjctcel en vhoters =(c) Cn twe vhoters i

ilj b

irh rhprhshlthj by EI ilj EB , tdhl tdhcr suf bi

cs i vhoter

rhprhshlthj by EO , wdhrh EO cs tdh jcikelia en tdh piriaahaekrif EIOB.


3/26

FIZD_

"filcsdgufirpdyscos.cl" ?

(cc)

i b b i (oeffutitcvh) (ccc) ( ) ( ) i b o i b o (isseocitcvh)

(cv)

i i i 0 0 (v)

i i i i ( ) ( )0

(vc) |b||i||bi|

(vcc) ||b||i|||bi|

Hxifpah # ? = Cn g?m2ci

ilj g8m:c2b

rhprhshlt twe ijmiohlt scjhs en i piriaahaekrif, nclj ulct

vhoters piriaaha te tdh jcikelias en tdh piriaahaekrif.

_eautcel = Aht IBOJ bh i piriaahaekrif suod tdit IB 6 i

ilj BO 6 b

.

Zdhl, IB + BO 6 IO

IO 6 bi

6 g2m>c?

|IO| 6 :?>3 6 ;

IB + BJ 6 IJ

BJ 6 IBIJ 6 ib

6 g

4m

2c

|BJ| 6 >::9 6 >3

Xlct vhoter iaelk IO 6 |IO|

IO6

;

9 g2m>c?

ilj Xlct vhoter iaelk BJ 6|BJ|

BJ6

>3

9 g4m2c

Hxifpah # : = IBOJH cs i phltikel. Prevh tdit tdh rhsuatilt en tdh nerohs IB, IH, BO , JO, HJ ilj IO

cs ?IO .

_eautcel = Aht R

bh tdh rhsuatilt neroh

R

6 IB + IH + BO + JO + HJ + IO

R

6 (IB + BO ) + (IH + HJ + JO) + IO

R

6 IO + IO + IO

R

6 ? IO . Dhloh prevhj.

Pesctcel vhoter en i peclt=Aht E bh i ncxhj erckcl, tdhl tdh pesctcel vhoter en i peclt P cs tdh vhoter EP . Cn i

ilj b

irh pesctcel

vhoters en twe peclts I ilj B, tdhl

IB 6 ib

6 pesctcel vhoter(p.v.)en B pesctcel vhoter(p.v.).) en I.

JC_ZILOH NERFXAIJcstiloh bhtwhhl tdh twe peclts I )i(

ilj B )b(

cs IB 6 bi

_HOZCEL NERFXAI

Cn i ilj b irh tdh pesctcel vhoters en twe peclts I ilj B, tdhl tdh p.v. en


4/26


5/26

FIZD_


_han Priotcoh Prebahfs

(:) Hxprhss v ho ters BO , OI ilj IB cl thrfs en tdh vhoters EI, EB ilj EO

(8) Cn b,i

irh pesctcel vhoters en tdh peclts (9, 9), (2, f), nclj tdh viauh en f ner wdcod i

ilj b

irh oeaaclhir.

(>) Zdh pesctcel vhoters en tdh peclts I, B, O, J irh g

m

c

, m

8c

2 , g

?m

2c

? , g

m

>c

rhsphotcvhay. _dew tdit tdh aclhs IB ilj OJ irh piriaaha ilj nclj tdh ritce en tdhcr ahlktds.

(;) Zdh vhrtcohs P, Y ilj _ en iPY_ divh pesctcel vhoters q,p

ilj s

rhsphotcvhay..

(c) Cn F cs tdh fcj peclt en PY, tdhl nclj pesctcel vhoter en F cl thrfs en qiljp

(cc) Nclj t

, tdh pesctcel vhoter en Z el _F suod tdit _Z = ZF 6 2 = 9, cl thrfs en q,p

ilj s

.

(ccc) Cn tdh piriaahaekrif PYR_ cs lew oefpahthj. Hxprhss r

, tdh pesctcel vhoter en tdh

peclt R cl thrfs en q,p

ilj s

(4) J, H, N irh tdh fcj-peclts en tdh scjhs BO, OI, IB rhsphotcvhay en i trcilkah.

_dew NH 62

9BO ilj tdit tdh suf en tdh vhoters IJ, BH , ON cs zhre.

(3) Zdh fhjcil IJ en i IBO cs bcs ho thj it H ilj BH cs preju ohj te fhht tdh scjh IO

cl N. _dew tdit IN 6?

9IO ilj HN 6

:

9BN..

(90) Peclt A, F, L jcvcjh tdh scjhs BO, OI, IB enIBO cl tdh ritces 9 = :, ? = 2, ? = ; rhsphotcvhay.

Prevh tdit IA + BF + OL cs i vhoter piriaaha te OG, wdhl G jcvcjhs IB cl tdh ritce 9 = ?.Ilswhrs = (:) EBEOBO , EOEIOI , EIEBIB (8) f 6 2

(>) 9 = 2 (;) f

62

9)qp(

, t

6

2

9)sqp(

, r

6

2

9)spq(

Ilkah bhtwhhl twe vhoters =Ct cs tdh sfiaahr ilkah nerfhj wdhl tdh clctcia peclts er tdh thrfclia peclts en tdh twe vhoters irh

breukdt tekhtdhr. Leth tdit 0 940 .

^hoter hquitcel en i aclh =Pirifhtrco vhoter hquitcel en i aclh pissclk tdreukd twe peclt )i(I

ilj )b(B

cs kcvhl by r

6 )ib(ti

,

wdhrh 't' cs i pirifhthr. Cn tdh aclh pisshs tdreukd tdh peclt I i( )

ilj cs piriaaha te tdh vhoter b

, tdhl

cts hquitcel cs btir

. Leth = r

cs tdh p.v. en tdh peclt el tdh aclh.


6/26

FIZD_

"filcsdgufirpdyscos.cl" >

I vhoter cl tdh jcrhotcel en tdh bcshoter en tdh ilkah bhtwhhl tdh twe vhoters bilji

csb

b

i

i

.

Dhloh bcshoter en tdh ilkah bhtwhhl tdh twe vhoters i

ilj b

cs i b , wdhrh R +. Bcshoter

en tdh hxthrcer ilkah bhtwhhl i

ilj b

cs i b , R +.

Lethtdit tdh hquitcels en tdh bcshoters en tdh ilkahs bhtwhhl tdh aclhs

r 6

i +

b ilj

r 6

i +

oirh =

r 6

i + t b o ilj r 6i + p o b .

_oiair prejuot Jet Prejuot en twe vhoters =Khefhtr coia clthrprhtit cel en soi air prejuot =Aht i

ilj b

bh vhoters rhprhshlthj by EI ilj EB rhsphotcvhay. Aht bh tdh ilkah bhtwhhl EI ilj

EB . Jriw BA EI ilj IF EB.

NrefEBA iljEIF, wh divh EA 6 EB oes ilj EF 6 EI oes.

(b)

(i)

Dhrh EA ilj EF irh glewl is premhotcels en b

el i

ilj i

el b

rhsphotcvhay..

Lew, b.i

6 | i

| | b

| oes 6 | i

|(| b

| oes )

6 | i

| (EB oes ) 6 | i

| (EA)

6 (Fiklctujh en i

) (Premhotcel en b

el i

) ........(c)

Ikicl b.i

6 | i

| | b

| oes 6 | b

| (| i

| oes )

6 | b

| (EI oes) 6 | b

| (EF)

6 (fiklctujh en b ) (Premhotcel en i el b ) ........(cc)

Zdus khefhtrcoiaay clthrprhthj, tdh soiair prejuot en twe vhoters cs tdh prejuot en fejuaus en hctdhr

vhoter ilj tdh premhotcel en tdh etdhr cl cts jcrhotcel.

(c) c .c 6 m .m 6 g . g 6 97 c .m 6 m . g 6 g .c 6 0

( cc) Premhotc el en |b|

b.ibeli

(ccc) Cn i

6 i9c + i

2m + i

?g ilj b

6 b

9c + b

2m + b

?g , tdhl b.i

6 i

9b

9+ i

2b

2+ i

?b

?

2

?

2

2

2

9 iiii

,

2

?

2

2

2

9

bbbb

(cv) Z dh ilkahbhtwhhl i

ilj b

cs kcvhl by|b||i|

b.ioes

, 0

(v) )0(,oesbib.i

leth tdit cn cs iouth, tdhl b.i

< 0 ilj cn cs ebtush, tdhl b.i

1 0

(vc) bi

6 oes|b||i|2|b||i| 22

, wdhrh cs tdh ilkah bhtwhhl tdh vhoters

(vcc) 22

iii.i

(vccc) i.bb.i

(oeffutitcvh)


7/26

FIZD_

"filcsdgufirpdyscos.cl" ;

(cx) o.ib.i)ob(.i

(jcstrcbutcvh)

(x) bi0b.i

)0b,0i(

(xc) )if(

. b

6 i

. )bf(

6 f )b.i(

(isseocitcvh), wdhrh f cs i soiair..

Leth=

(i) Fix cfuf v iauh en

i .

b cs

i

b (b) Fclcfuf v iauh en

i .

b cs

i

b

(o) Ily vhoter i oil bh wrctthl is

i 6

i c c i m m i g g . . . .

Hxifpah # ; = Nclj tdh viauh en p ner wdcod tdh vhoters g3m2c?i

ilj g?mpcb

irh

(c) phrphljcouair (cc) piriaaha

_eautcel = (c) bi

b.i

6 0 g3m2c? . g?mpc 6 0 ? + 2p + 2; 6 0 p 6 98

(cc) vhoters i 6 g3m2c? ilj b 6 g?mpc irh piriaaha cnn

9

?6

p

26

?

3 ? 6

p

2 p 6

?

2

Hxifpah # 4 = Cn i

+ b

+ o

6 0

, |i|

6 ?, |b|

6 8 ilj |o|

6 ;, nclj tdh ilkah bhtwhhl i

ilj b

.

_eautcel = [h divh, 0obi

bi

6 o bi

. bi

6 o

. o

2bi 6 2|o| 2i + 2b + b.i2 6 2o

2

i

+2

b

+ 2 i

b

oes 6 2o

3 + 28 + 2 (?) (8) oes 6 :3 oes 62

9 6

?

.

Hxifpah # 3 = Nclj tdh viauhs en x ner wdcod tdh ilkah bhtwhhl tdh vhoters i

6 2x2 c + :x m + g ilj b

6 ; c

2m + x g cs ebtush.

_eautcel = Zdh ilkah bhtwhhl vhoters i

ilj b

cs kcvhl by oes 6|b||i|

b.i

Lew, cs ebtush oes 1 0 |b||i|

b.i

1 0

b.i

1 0 W 0|b||,i|

V

9:x2 4x + x 1 0 ;x (2x 9) 1 0

x(2x 9) 1 0 0 1 x 12

9

Dhloh, tdh ilkah bhtwhhl tdh kcvhl vhoters cs ebtush cn x (0, 9/2)


8/26

FIZD_


Hxifpah#90 = J cs tdh fcj peclt en tdh scjh BO en iIBO, sdew tdit IB2 + IO2 6 2 (IJ2 + BJ2)

_eautcel = [h divh IB 6 IJ + JB

IB2 6 2)JBIJ(

IB2 6 IJ2 + JB2 + IJ2 . JB .........(c)

Iase wh divh IO 6 IJ + JO

IO2 6 2)JOIJ(

IO2 6 IJ2 + JO2 + IJ2 . JO ........(cc)

Ijjclk (c) ilj (cc) , wh kht IB2 + IO2 6 2IJ2 + 2BJ2 + 2IJ . )JOJB(

IB2 + IO2 6 2(IJ2 + BJ2) JB + JO 6 0

Hxifpah#99= Cn i

6 c + m + g ilj b

6 2c m + ? g , tdhl nclj

(c) Oef pelhlt en b

iaelk i

. (cc) Oefpelhlt en b

cl pailh en b&i

but te i

.

_eautcel = (c) Oef pelhlt en b

iaelk i

cs

2|i|

b.i

i

Dhrh i

. b

6 2 9 + ? 6 :

2|i|

6 ?

Dhloh

2|i|

b.i

i

6?

:i

6?

:(c + m + g )

(cc) Oef pelhlt en b

cl pailh en b&i

but te i

cs b

2|i|

b.i

i

. 6?

9 g8m;c2


(99) Cn i

ilj b

irh ulct vhoters ilj cs ilkah bhtwhhl tdhf, prevh tdit til2

6

|bi|

|bi|

.

(92) Nclj tdh viauhs en x ilj y cn tdh vhoters i

6 gmxc? ilj b

6 gymc2 irh futuiaay

phrphljcouair vhoters en hquia fiklctujh.

(9?) Aht i

6 g2m2cx2 , b

6 gmc ilj o

6 g:m8cx2 bh tdrhh vhoters. Nclj tdh viauhs en x

ner wdcod tdh ilkah bhtwhhl i

ilj b

cs iouth ilj tdh ilkah bhtwhhl b

ilj o

cs ebtush.

(9:) Zdh peclts E, I, B, O, J irh suod tdit iEI

, bEB

, b?i2EO

, b2iEJ

.

Kcvhl tdit tdh ahlktd en EI cs tdrhh tcfhs tdh ahlktd en EB . _dew tdit BJ ilj IO irh

phrphljcouair.

(98) IBOJ cs i thtridhjrel ilj K cs tdh ohltrecj en tdh bish BOJ. Prevh tdit

IB2 + IO2 + IJ2 6 KB2 + KO2 + KJ2 + ?KI2


9/26

FIZD_


Ilswhrs = (92) x 6 92

?9, y 6

92

:9(9?) ( ?, 2) (2, ?)

^hoter prejuot Oress Prejuot e n twe vhoters =(c) Cn b,i

irh twe vhoters ilj cs tdh ilkah bhtwhhl tdhf, tdhl lsclbibxi

, wdhrh l cs tdh

ulct vhoter phrphljcouair te betd bilji suod tdit liljb,i nerfs i rckdt diljhj sorhw systhf.

(cc) K hefhtrco ia ay bxi

6 irhi en tdh piriaahaekrif wdesh twe ijmiohlt scjhs irh rhprhshlthj by

bilji

.

(ccc) 0ggmmcc 7 mcg,cgm,gmc

(cv) Cn i

6 i9c +i

2m + i

? g ilj b

6 b

9c + b

2m + b

? g , tdhl

?29

?29

bbb

iii

gmc

bi

(v)

i x b b x i (let oeffutitcvh)

(vc) ( )f i

b 6i ( )f b

6 f ( )

i b (isseocitcvh), wdhrh f cs i soiair..

(vcc) i x b o i x b i x o( ) ( ) ( ) (jcstrcbutcvh)

(vccc) bilji0bi

irh piriaaha (oeaaclhir) ( , ) i b 0 0 c.h. i G b , wdhrh G cs i soiair..

(cx) Xlct vhoter phrphljcouair te tdh pailh en i

ilj b

cs l 6|bi|

bi

(x) I vhoter en fiklctujh r ilj phrphljcouair te tdh pailh en i

ilj b

cs|bi|

)bi(r

(xc) Cn cs tdh ilkah bhtwhhlbi

bxiscltdhl,bilji

(xcc) Cn oiljb,i

irh tdh pesctcel vhoters en ? peclts I, B ilj O rhsphotcvhay, tdhl tdh vhoter irhi en

IBO 6 9

2

i x b b x o ox i . Zdh peclts I, B ilj O irh oeaaclhir cn 0ixooxbbxi

(xccc) Irhi en ily quijrcaithria wdesh jcikelia vhoters irh29 jiljj

cs kcvhl by 29 jxj

2

9

(xcv) Aikrilkh's Cjhltcty = Ner ily twe vhotersb.bb.i

b.ii.i)b.i(bi)bxi(7bilji 2

222

Hxifpah#92 = Nclj i vhoter en fiklctujh 3, wdcod cs phrphljcouair te betd tdh vhoters g?mc:

ilj g2mc2 .


10/26

FIZD_


_eautcel = Aht i

6 g?mc: ilj b

6 g2mc2 . Zdhl

bi

6292

?9:

gmc

6 (2 ?) c (4 + >) m + (: 2) g 6 g2m2c

|bi|

6 222

22)9( 6 ?

Rhqucrhj vhoter 6 3

|bi|

bi

6?

3)g2m2c( 6 ( g>m>c? )

Hxifpah#9? = Ner ily tdrhh vhoters o,b,i

, sdew tdit 0)bi(o)io(b)ob(i

.

_eautcel = [h divh, i

)ob(

+ b

)io(

+ o

)bi(

6 boioiboboibi

WXsclk jcstrcbutcvh aiwV

6 oboibioboibi

6 0

W biib

htoV

Hxifpah#9: = Ner ily vhoter i

, prevh tdit 2|ci|

+ 2|mi|

+ 2|gi|

6 2 2|i|

_eautcel = Aht i

6 gimici ?29 . Zdhl

ci

6 )gimici( ?29 c 6 i9 )cc( + i2 )cm( + i? )cg( 6 i2 mig ?

2|ci|

6 i2

2 + i?

2

mi

6 )gimici( ?29 m 6 cigi ?9

2|mi|

6 i29

+ i?

2

gi

)gimici( ?2c g

6 cimi 2c

2|gi|

6 i9

2 + i2

2

2|ci|

+ 2|mi|

+ 2|gi|

6 i2

2 + i?

? + i9

2 + i?

2 + i9

2 + i2

2

6 2 (i9

2 + i2

2 + i?

2) 6 2 2|i|

Hxifpah#98 = Aht EI 6 i

, EB 6 90 i

+ b2

ilj EO 6 b

wdhrh E cs erckcl. Aht p jhleth tdh irhi en tdh

quijrcaithria EIBO ilj q jhleth tdh irhi en tdh piriaahaekrif wctd EI ilj EO is ijmiohlt

scjhs. Prevh tdit p 6 >q._eautcel = [h divh,

p 6 Irhi en tdh quijrcaithria EIBO

p 62

9|IOEB| 6

2

9|)EIEO(EB|

p 62

9|)ib()b2i90(|

6

2

9|)ib(2)bb(2)ii(90)bi(90|

p 62

9|)bi(200)bi(90|

6 > |bi|

.....(c)

ilj q 6 Irhi en tdh piriaahaekrif wctd EI ilj EO is ijmiohlt scjhs

q 6 |EOEI| 6 |bi|

........(cc)

N ref (c) ilj (cc) , wh kht p 6 >q


11/26

FIZD_



(9>) Cn p

ilj q

irh ulct vhoters nerfclk il ilkah en ?0. Nclj tdh irhi en tdh piriaahaekrif divclk

q2pi

ilj qp2b

is cts jcikelias.

(9;) Prevh tdit tdh lerfia te tdh pailh oelticlclk tdh tdrhh peclts wdesh pesctcel vhoters irh

o,b,i

achs cl tdh jcrhotcel biioob

(94) IBO cs i trcilkah ilj HN cs ily strickdt aclh piriaaha te BO fhhtclk IO, IB cl H, N rhsphotcvhay.

Cn BR ilj OY bh jriwl piriaaha te IO, IB rhsphotcvhay te fhht HN cl R ilj Y rhsphotcvhay,

prevh tdit IRB 6IOY.

Ilswhrs = (9>) ?/: sq. ulcts

_derthst jcstiloh bhtwhhl twe aclhs =Cn twe aclhs cl spioh clthrshot it i peclt, tdhl ebvceusay tdh sderthstjcstiloh bhtwhhl tdhf cs zhre. Aclhs wdcod je let clthrshot ilj irh

iase let piriaaha irh oiaahj sghw aclh. Ner _ghw aclhs tdh jcrhotcel en

tdh sderthst jcstiloh weuaj bh phrphljcouair te betd tdh aclhs.

Aht AF bh tdh sderthst jcstiloh vhoter bhtwhhl tdh aclhs A9ilj A2. Zdhl AF cs phrphljcouair te betd

p

ilj q

c.h. AF cs piriaaha te p

q

. Zdhrhnerh tdh fiklctujh en tdh sderthst jcstiloh vhoter (c.h.

| AF|) weuaj bh hquia te tdit en tdh premhotcel en IB iaelk tdh jcrhotcel en tdh aclh en sderthst jcstiloh.

AFelIBenemhotcelPr|AF| 6 qxpelIBenemhotcelPr

6qxp

)qxp(.)ib(

qxp

)qxp(.IB

(c) Zdh twe aclhs jcrhothj iaelk qiljp

wcaa clthrshot elay cn sderthst jcstiloh 6 0

c.h. 0)qxp(.)ib(

c.h. ib

achs cl tdh pailh oelticlclk qiljp

. 0qpib

.

(cc) Cn twe piriaaha aclhs irh kcvhl by bGiriljbGir 2299

, tdhl jcstiloh (j) bhtwhhl tdhf cs

kcvhl byb

)ii(xbj 92

_oi air t rcpah prejuot Bex Prejuot _ .Z.P. =(c) Zdh soiair trcpah prejuot en tdrhh vhoters oiljb,i

cs jhnclhj is= obio.bxi

oes.scl ,

wdhrh cs tdh ilkah bhtwhhl b,i

(c.h.

) ilj cs tdh ilkah bhtwhhl oiljbxi

(c.h. o 6 ) . Ct cs (c.h. o.bi ) iase wrctthl is obi ilj sphaahj is bex prejuot.(cc) _oiair trcpah prejuot khefhtrcoiaay rhprhshlts tdh veaufh

en tdh piriaahaepcphj wdesh tdrhh oethrfcleus hjkhs irh

rhprhshlthj by .h.coiljb,i ^ 6 VobiW


12/26

FIZD_


(ccc) Cl i soiair trcpah prejuot tdh pesctcel en jet ilj oress oil bh clthrodilkhj c.h.

VbioWViobWVobiWo.)bxi()oxb(.i

(cv) VboiWVobiW.h.c)bxo(.i)oxb(.i

(v) Cn i

6 i9c + i

2m + i

?g 7 b

6 b9c +b

2m +b

?g ilj o

6 o

9c + o

2m + o

?g , tdhl

?29

?29

?29

ooo

bbb

iii

VobiW

.

Cl khlhria, cn lifiii ?29

7 lbfbbb ?29

ilj lofooo ?29

tdhl lfooo

bbb

iii

obi

?29

?29

?29

, wdhrh liljf,

irh lel-oepailir vhoters.

(vc) Cn oiljb,i

irh oepailir 0VobiW

.

(vcc) _oiair prejuot en tdrhh vhoters, twe en wdcod irh hquia er piriaaha cs 0 0VobiW

,

(vccc) Cn i b o, , irh lel-oepailir, tdhl 0VobiW

ner rckdt diljhj systhf ilj 0VobiW

ner ahnt diljhj

systhf.

(cx) VgmcW 6 9 (x) VobiWGVobiGW

(xc) VjobWVjoiWVjo)bi(W

(xcc) i b b o o i 6 0 ilj

i b b o o i 6 2 obi

.

(xccc) 2obi

6

o.ob.oi.o

o.bb.bi.b

o.ib.ii.i

Zhtridhj rel ilj cts prephr tc hs =(i) Zdh veaufh en tdh thtridhjrel EIBO wctd E is erckcl ilj tdh pesctcel vhoters en I, B ilj O bhclk

oiljb,i

rhsphotcvhay cs kcvhl by>

9^ obi

(b) Cn tdh pesctcel vhoters en tdh vhrtcohs en thtridhjrel irh jiljo,b,i

, tdhl tdh pesctcel vhoter en cts

ohltrecj cs kcvhl by )jobi(:

9 .

leth tdit tdcs cs iase tdh peclt en oelourrhloy en tdh aclhs meclclk tdh vhrtcohs te tdh ohltrecjs en tdh

eppescth niohs ilj cs iase oiaahj tdh ohltrh en tdh thtridhjrel. Cl oish tdh thtridhjrel cs rhkuair ct cshqucjcstilt nref tdh vhrtcohs ilj tdh neur niohs en tdh thtridhjrel.

Hxifpah#9> = Nclj tdh veaufh en i piriaahaepcphj wdesh scjhs irh kcvhl by g8m;c? , g?m;c8

ilj g?m8c;

_eautcel = Aht g8m;c?i

, g?m;c8b

ilj g?m8c;o

.

[h glew tdit tdh veaufh en i piriaahaepcphj wdesh tdrhh ijmiohlt hjkhs irh o,b,i

cs

VobiW

.


13/26

FIZD_

"filcsdgufirpdyscos.cl" 9?

Lew VobiW

6?8;

?;8

8;?

6 ? (29 98) ; (98 + 29) + 8 (28 :3)

6 904 282 920 6 2>:

_e rhqucrhj veaufh en tdh piriaahaepcphj 6 VobiW

6 | 2>: | 6 2>: oubco ulcts

Hxifpah#9; = _cfpacny VioobbiW

_eautcel = [h divh =

VioobbiW

6 )}ob()bi{(

. )io(

WBy jhnclctcelV

6 )obbboibi(

. )io(

WBy jcstrcbutcel aiwV

6 )obiobi(

. )io(

W 0bb

V

6 )bi(

. o

)bi(

. i

+ )io(

. o

)io(

. i

+ )ob(

. o

)ob(

. i

WBy jcstrcbutcel aiwV

6 VobiW

VibiW

+ VoioW

ViioW

+ VoobW

ViobW

6 VobiW

ViobW

W[dhl ily twe vhoters irh hquia, soiair trcpah prejuot cs zhre V

6 VobiW

VobiW

6 0 W ViobW

6 VobiW

V

Hxifpah#94 = Nclj tdh veaufh en tdh thtridhjrel wdesh neur vhrtcohs divh pesctcel vhoters i

, b

, o

ilj j

.

_eautcel = Aht neur vhrtcohs bh I, B, O, J wctd pesctcel vhoters i

, b

, o

ilj j

rhsphotcvhay..

JI 6 ( i

j

)

JB

6 ( b

j

)

JO 6 ( o

j

)

Dhloh veaufh ^ 6>

9W i

j

b

j

o

j

V

6>

9( i

j

) . W( b

j

) ( o

j

)V

6>

9( i

j

) . W b

o

b

j

+ o

j

V

6 >

9

{W i

b

o

V W i

b

j

V + W i

o

j

V W j

b

o

V}

6>

9{W i

b

o

V W i

b

j

V + W i

o

j

V W b

o

j

V}

Hxifpah#93 = _dew tdit tdh vhoters g2m2c:b,g2m:c2i

ilj g:m2c2o

irh oepailir..

_eautcel = VobiW

6

:22

22:

2:2

6 2(4 :) :(9> :) 2(4 :)

6 2: :4 + 2: 6 0_e vhoters i

, b

, o

irh oepailir


14/26

FIZD_

"filcsdgufirpdyscos.cl" 9:


(93) _dew tdit {( i

+ b

+ o

) ( o

b

)} . i

6 2 obi

.

(20) _dew tdit 0)obi()ob(.i

(29) Elh vhrthx en i piriaahaepcphj cs it tdh peclt I (9, 9, 2) cl tdh rhotilkuair oirthscil oe-erjclith. Cn

tdrhh ijmiohlt vhrtcohs irh it B(9, 0, 2), O(2, 2, ?) ilj J(:, 2, 9), tdhl nclj tdh veaufh en tdhpiriaahaepcphj.

(22) Nclj tdh viauh en f suod tdit tdh vhoters gmc2 , g?m2c ilj g8mfc? irh oepailir..

(2?) _dew tdit tdh vhoter o,b,i

irh oepailir cn ilj elay cn ob

, io

, bi

irh oepailir..

Ilswhrs = (29) ;2 (22) :

^hoter trcpah prejuot =Aht oiljb,i

bh ily tdrhh vhoters, tdhl tdh hxprhsscel )oxb(xi

cs i vhoter & cs oiaahj i vhoter

trcpah prejuot.

Khefhtrcoia clthrprhtitcel en )oxb(xi

Oelscjhr tdh hxprhsscel i x b x o( ) wdcod ctshan cs i vhoter, scloh ct cs i oress prejuot en twe vhoters

)oxb(ilji

. Lew i x b x o( ) cs i vhoter phrphljcouair te tdh pailh oelticlclk )oxb(ilji

but

b x o

cs i vhoter phrphljcouair te tdh pailh oelticlclk oiljb

, tdhrhnerh i x b x o( ) cs i vhoter wdcod achs

cl tdh pailh en oiljb

ilj phrphljcouair tei . Dhloh wh oil hxprhss )oxb(xi

cl thrfs en oiljb

c.h. )oxb(xi

6 oybx

, wdhrh x , y irh soiairs.

i x b x o( ) 6 ( . ) ( . )

i o b i b o ( )

i x b x o 6 ( . ) ( . )

i o b b o i

Cl khlhria ( ) ( ) i x b x o i x b x o

Hxifpah#20 = Ner ily vhoter i

, prevh tdit )gi(g)mi(m)ci(c

6 i2

_eautcel = Aht gimicii ?29

.

Zdhl )gi(g)mi(m)ci(c

6 }c)i.c(i)c.c({

+ }m)i.m(i)m.m{(

+ }g)i.g(i)g.g{(

6 }m)i.m(i{}c)i.c(i{(

+ }g)i.g(i{

6 g)i.g(m)i.m(c)i.c{(i?

6 )gimici(i? ?29

6 i2ii?

Hxifpah#29 =

Prevh tdit )}jo(b{i

6 )oi)(j.b(

)o.b(

)ji(

_eautcel = [h divh, )}jo(b{i

6 }j)o.b(o)j.b{(i

6 }j)o.b{(i}o)j.b{(i

Wby jcst. aiwV

6 )ji()o.b()oi()j.b(


15/26

FIZD_


Hxifpah#22 = Aht i

6 c + m2 g? , b

6 c + m2 g2 ilj o

6 c2 m + g. Nclj tdh viauh(s) en, cn ily,,

suod tdit obbi

io

6 0

.

_eautcel = obbi

io

6 obi

b

io

6 obi io.bob.i

wdcod vilcsdhs cn (c) b.i

o

6 o.b

i

(cc) obi

6 0

(c) b.i

o

6 o.b

i

ahijs te tdh hquitcel 2? + 90 + 92 6 0,2 + > 6 0 ilj > > 6 0,

wdcod je let divh i oeffel seautcel.

(cc) obi

6 0

92

229

?2

6 0 ?6 2 6?

2

Hxifpah#2? = C n BI

6 i

, I

. i

6 9 ilj BI

6 b

, tdhl prevh tdit I

6 2|i|

ibi

ilj

B

6

2

2

|i|

9|i|iib

.

_eautcel = Kcvhl iBI

.....(c)

BI.i

6 i.i

iI.i

. B

6 i.i

9 + B.i

6 2|i|

B.i

6 2|i|

9 .......... .(cc)

Kcvhl bBI

BIi

6 i

b

B.i

I

I.i

biB

9|i| 2

BI

6 bi

.. ... ... ...( ccc) WXsclk hquitcel (cc)V

seavclk hquitcel (c) ilj (ccc) scfuatilheusay, wh kht

I

6 2|i|

ibi

ilj B

6

2

2

|i|

9|i|iib

Hxifpah#2: = _eavh ner r

sitcsnyclk tdh scfuatilheus hquitcels bobr

, 0i.r

prevcjhj i

cs let

phrphljcouair teb

.

_eautcel = )or(

b

6 0

or

ilj b

irh oeaaclhir

bgor

r 6 bgo ........(c)

i.r

6 0


16/26

FIZD_

"filcsdgufirpdyscos.cl" 9>

)bgo(

. i

6 0

g 6 b.i

o.i

puttclk cl (c) wh khtb.i

o.ior

b

Hxifpah#28 = C n bxgix

, wdhrh g cs i soiair ilj b,i

irh ily twe vhoters, tdhl jhthrfclh x

cl thrfs

en b,i ilj g.

_eautcel = bxgix

..........(c)

Prhfuatcpay tdh kcvhl hquitcel vhoterciaay by i

)ix(i

+ g )xi(

6 bi

bi)xi(gi)x.i(x)i.i(

..........(cc)

Prhfuatcpay (c) soiairay by i

VixiW

+ )x.i(g

6 b.i

b.i)x.i(g

.......(ccc)

_ubstctutclk ix

nref (c) ilj x.i

nref (ccc) cl (cc) wh kht

x

6 22 gi

9

i

g

)b.i()bi(bg


(2:) Prevh tdit 0)bi(o)io(b)ob(i

.

(28) Nclj tdh ulct vhoter oepailir wctd c + m + 2 g ilj c + 2m + g ilj phrphljcouair te c + m + g .

(2>) Prevh tdit )ib()i.i()}bi(i{i

.

(2;) Kcvhl tdit2p

9x

)x.p(

qp

, sdew tdit q.p2

9x.p

ilj nclj x

cl thrfs en p

ilj q

.

(24) Cn i.x

6 0, b.x

6 0 ilj o.x

6 0 ner sefh lel-zhre vhoter x

, tdhl sdew tdit VobiW

6 0

(23) Prevh tdit r

6 VobiW

)ob()i.r(

+ VobiW

)io()b.r(

+ VobiW

)bi()o.r(

wdhrh o,b,i

irh tdrhh lel-oepailir vhoters

Ilswhrs = (28) 2

9( m + g ) ilj x

6 q

p

|p|2

q.p2

A clhir oefbclitcels =Kcvhl i nclcth sht en vhoters

i b o, , , ...... , tdhl tdh vhoter

r xi yb zo ........ cs oiaahj i aclhir

oefbclitcel en

i b o, , , ...... ner ily x, y, z.....R. [h divh tdh neaaewclk rhsuats=

(i) Cn i b, irh lel zhre, leloeaaclhir vhoters, tdhl 'yy,'xxb'yi'xbyix


17/26

FIZD_

"filcsdgufirpdyscos.cl" 9;

(b) Nuljifhltia Zdherhf cl pailh = Aht i b, bh lel zhre, lel oeaaclhir vhoters, tdhl ily vhoter

r

oepailir

wctd i b, oil bh hxprhsshj ulcquhay is i aclhir oefbclitcel en bilji

c.h. tdhrh hxcst sefh ulcquh x, y R suod tdit xi yb r

.

(o) Cn

i b o, , irh lelzhre, leloepailir vhoters, tdhlxi yb zo x i y b z o x x y y z z

' ' ' ' , ' , '

(j) Nuljifhltia tdherhf cl spioh=Aht i b o, , bh lelzhre, leloepailir vhoters cl spioh.

Zdhl ily vhoterr oil bh ulcquhay hxprhsshj is i aclhir oefbclitcel en

i b o, , c.h. tdhrh hxcst

sefh ulcquh x,y, z R suod tdit xi yb zo r

.

(h) C n l29 x......,,x,x

i r h l l e l z h r e v h o t e r s i l j g9,g

2,.....,g

l i r h l s o i a i r s i l j c n t d h a c l h i r

o efbc lit c el 0g.....,,0g,0g0xg.......xgxgl29ll2299

, t d h l w h s i y t d i t v h o t e r s

l29 x......,,x,x

irh aclhiray cljhphljhlt vhoters.

(n) Cn l29 x......,,x,x

irh letaclhiray cljhphljhlttdhl tdhy irh sicj te bhaclhiray jhphljhlt vhoters.

c .h. c n 0xg......xg......xgxgxg llrr??2299

ilj cn tdhrh hxcsts it ahist elh g r 0, tdhl

l29 x......,,x,x

irh sicj te bh a clhiray jhphljhlt vhoters.

Leth 9= Cn gr 07 0xg......xg......xgxgxg llrr??2299

g x g x g x g x g x g xr r r r r r l l

9 9 2 2 9 9 9 9....... . . . .....

gg

x gg

x gg

x gg

x gg

xrr

r

r r

r

r

r l

r

l

9 9 9 9 99 9 2 2 9 9

..... . .....

9r9r2299r xo......xoxox

+ ll9r9r xo.......xo

c.h.x r cs hxprhsshj is i aclhir oefbclitcel en vhoters l9r9r29 x,..........,x,x,........,x,x

Dhloh l9r9r29r x.......,,x,x........,,x,xwctdx

nerfs i aclhiray jhphljhlt sht en vhoters.

Leth 2=

Cni 6 ?c + 2m + 8 g tdhl

i cs hxprhsshj is i Aclhir Oefbclitcelen vhoters c , m , g . IaseIase

i , c , m ,

g nerf i aclhiray jhphljhlt sht en vhoters. Cl khlhria, cl ? jcfhlscelia spioh hvhry sht en neur vhoterscs i aclhiray jhphljhlt systhf.

c ,m , g irh Aclhiray Cljhphljhlt sht en vhoters. Ner

G9c + G

2m + G

?g 6 0

G

96 G

26 G

?6 0

Zwe vhoters i

ilj b

irh aclhiray jhphljhlti cs piriaaha te

b c.h. 0bi

aclhir jhphljhloh en i

ilj b

. Oelvhrshay cn 0bi

tdhl i

ilj b

irh aclhiray cljhphljhlt.

Cn tdrhh vhoters i b o, , irh aclhiray jhphljhlt, tdhl tdhy irh oepailir c.h. VobiW

6 0. Oelvhrshay cn

VobiW

0 tdhl tdh vhoters irh aclhiray cljhphljhlt.


18/26

FIZD_


Hxifpah#2> = Kcvhl tdit pesctcel vhoters en peclts I, B, O irh rhsphotcvhay

i

2 b

+ ? o

, 2 i

+ ? b

: o

, ; b

+ 90 o

tdhl prevh tdit vhoters IB ilj IO irh aclhiray

jhphljhlt.

_eautcel = Aht I, B, O bh tdh kcvhl peclts ilj E bh tdh peclt en rhnhrhloh tdhl

EI 6 i

2 b

+ ? o

, EB 6 2 i

+ ? b

: o

ilj EO 6 ; b

+ 90 o

Lew IB 6 p.v. en B p.v. en II

6 EB EI 6 ( i

+ 8 b

; o

)

ilj IO 6 p.v. en O p.v en II

6 EIEO 6 )o;b8i(

6 IB

IO 6 IB wdhrh6 9. Dhloh IB ilj IO irh aclhiray jhphljhlt

Hxifpah#2;= Prevh tdit tdh vhoters 8 i

+ > b

+ ; o

, ; i

4 b

+ 3 o

ilj ? i

+ 20 b

+ 8 o

irh aclhiray

jhphljhlt, wdhrh i

, b

, o

bhclk aclhiray cljhphljhlt vhoters.

_eautcel = [h glew tdit cn tdhsh vhoters irh aclhiray jhphljhlt , tdhl wh oil hxprhss elh en tdhf is i

aclhir oefbclitcel en tdh etdhr twe.

Lew aht us issufh tdit tdh kcvhl vhoter irh oepailir, tdhl wh oil wrcth

8 i

+ > b

+ ; o

6 ( ; i

4 b

+ 3 o

) + f (? i

+ 20 b

+ 8 o

)

wdhrh , f irh soiairs

Oefpirclk tdh oehnncochlts en i

, b

ilj o

el betd scjhs en tdh hquitcel

8 6 ; + ?f ..........(c)

> 6 4 + 20 f ..........(cc)

; 6 3 + 8f ..........(ccc)

Nref (c) ilj (ccc) wh kht

: 6 4 6 29

6 f wdcod hvcjhltay sitcsnchs (cc) hquitcel tee.

Dhloh tdh kcvhl vhoters irh aclhiray jhphljhlt .


(?0) Jehs tdhrh hxcst soiairs u, v, w suod tdit chwhvhu ?29

wdhrh gh9

, gmh2

,

g2mh?

5

(?9) Oelscjhr i bish o,b,i

ilj i vhoter ob?i2

. Oefputh tdh oe-erjcliths en tdcs vhoter

rhaitcvhay te tdh bish r,q,p

wdhrh b?i2p

, ob2iq

, o2bi?r

.

(?2) Cn i

ilj b

irh lel-oeaaclhir vhoters ilj I

6(x + :y) i

+ (2x + y + 9) b

ilj B

6 (y 2x + 2)

i

+ (2x ?y 9) b

, nclj x ilj y suod tdit B2I?

.

(??) Cn vhoters o,b,i

bh aclhiray cljhphljhlt, tdhl sdew tdit

(c) o?b2i

, o:b?i2

, o2b

irh aclhiray jhphljhlt

(cc) o2b?i

, ob:i2

, ob2i?

irh aclhiray cljhphljhlt.


19/26

FIZD_


(?:) K cvhl tdit mc , m2c irh twe vhoters. Nclj i ulct vhoter oepailir wctd tdhsh vhoters ilj

phrphljcouair te tdh ncrst vhoter mc . Nclj iase tdh ulct vhoter wdcod cs phrphljcouair te tdh

pailh en tdh twe kcvhl vhoters.

(?8) Cn wctd rhnhrhloh te i rckdt diljhj systhf en futuiaay phrphljcouair ulct vhoters g,m,c ,

m

c

?

, g

?m

c

2

. Hxprhss

cl tdh nerf 29

wdhrh 9

cs piriaaha te

ilj

2

cs phrphljcouair te

.

(?>) Prevh tdit i vhoter r

cl spioh oil bh hxprhsshj aclhiray cl thrfs en tdrhh lel-oepailir,,

lel-zhre vhoters o,b,i

cl tdh nerfVobiW

oVbirWbViorWiVobrWr

Ilswhrs = (?0) Le (?9) (0, ;/8, 9/8) (?2) x 6 2, y 6 9

(?:) 2

9)mc( 7 g (?8) m

2

9c

2

?9

, g?m2

?c

2

92

Rho cpreoia systhf en vhoters =Cn

i b o, , &

i ' , b ' , o ' irh twe shts en lel-oepailir vhoters suod tdit

i.i' 6 b.b' 6 o.o' 6 9 , tdhl tdh

twe systhfs irh oiaahj Rhocpreoia _ysthf en vhoters.

Leth =VobiW

bxioilj

VobiW

ixob,

VobiW

oxb6i

Hxifpah#24= Cn o,b,i

ilj o,b,i

bh tdh rhocpreoia systhf en vhoters, prevh tdit

(c) ?o.ob.bi.i

(cc) 0oobbii

_eautcel = (c) [h divh = i.i

6 b.b

6 o.o

6 9

i.i

+ b.b

+ o.o

6 9 + 9 + 9 6 ?

(cc) [ h divh = i

6 )ob(

, b

6 )io(

ilj o

6 )bi(

, wdhrh 6VobiW

9

}o)b.i(b)o.i{()}ob(i{)ob(iii

}i)o.b(o)i.b{()}io(b{)io(bbb

ilj }b)i.o(i)b.o{()}bi(o{)bi(ooo

oobbii

6 }b)i.o(i)b.o{(}i)o.b(o)i.b{(}o)b.i(b)o.i{(

6 Vb)i.o(i)b.o(i)o.b(o)i.b(o)b.i(b)o.iW(

6 Vb)o.i(i)o.b(i)o.b(o)b.i(o)b.i(b)o.iW(

6 00

Hquitcel en i pailh =(c) Zdh hquitcel 0l.)rr( 0

rhprhshlts i pailh oelticlclk tdh peclt wctd pesctcel vhoter

lwdhrh,r0

cs i vhoter lerfia te tdh pailh.


20/26

FIZD_


Zdh ibevh hquitcel oil iase bh wrctthl is jl.r

, wdhrh j 6 l.r0

(cc) Ilkah bhtwhhl twe pailhs cs tdh ilkah bhtwhhl twe lerfias jriwl te tdh pailhs ilj tdh ilkah

bhtwhhl i aclh ilj i pailh cs tdh oefpacfhlt en tdh ilkah bhtwhhl tdh aclh ilj tdh lerfia te

tdh pailh.

(ccc) Zdh ahlktd en phrphljcouair (p) nref i peclt divclk pesctcel vhoter i

te tdh pailh jl.r

cs

kcvhl by p 6 |l|

|jl.i|

(cv) Cn )ir(

. 0l9

ilj )ir(

. 0l2

irh tdh hquitcels en twe pailhs, tdhl tdh hquitcel en aclh

en clthrshotcel en tdhsh pailhs cs kcvhl by )ll(ir 29

.

Zhst en oeaaclhir cty =Zdrhh peclts I,B,O wctd pesctcel vhoters

i b o, , rhsphotcvhay irh oeaaclhir, cn & elay cn tdhrh hxcst

soiairs x, y, z let iaa zhre scfuatilheusay suod tdit xi yb zo

6 0

, wdhrh x + y + z 6 0.

Zhst en oepailircty =Neur peclts I, B, O, J wctd pesctcel vhoters

i b o j, , , rhsphotcvhay irh oepailir cn ilj elay cn tdhrh

hxcst soiairs x, y, z, w let iaa zhre scfuatilheusay suod tdit x i + y b + z o + w j

6 0

, wdhrh

x + y + z + w 6 0.

Hxifpah#23 = _dew tdit tdh vhoters o?bi2

, o2bi

ilj o?bi

irh lel-oepailir vhoters.

_eautcel = Aht, tdh kcvhl vhoters bh oepailir.

Zdhl elh en tdh kcvhl vhoters cs hxprhsscbah cl thrfs en tdh etdhr twe.Aht o?bi2

6 x o2bi

+ y o?bi

, ner sefh soiairs x ilj y..

o?bi2

6 (x + y) i

+ (x + y) b

+ (2x ?y) o

2 6 x + y, 9 6 x + y ilj ? 6 2x ?y._eavclk ncrst ilj tdcrj en tdhsh hquitcels, wh kht x 6 3 ilj y 6 ;.

Oahiray tdhsh viauhs je let sitcsny tdh shoelj hquitcel.

Dhloh tdh kcvhl vhoters irh let oepailir.

Hxifpah#?0= Prevh tdit neur peclts ob?i2

, o?b2i

, o2b:i?

ilj o>b>i

irh oepailir..

_eautcel = Aht tdh kcvhl neur peclts bh P, Y, R ilj _ rhsphotcvhay. Zdhsh peclts irh oepailir cn tdh vhoters

PY, PR ilj P_ irh oepailir. Zdhsh vhoters irh oepailir cnn elh en tdhf oil bh hxprhsshj

is i aclhir oefbclitcel en etdhr twe. _e aht PY 6 x PR + y P_

o:b8i

6 x obi

+ y o;b3i

o:b8i

6 (x y) i

+ (x 3y) b

+ (x + ;y) o

x y 6 9, x 3y 6 8, x + ;y 6 : WHquitclk oehnn. en o,b,i

el betd scjhsV

_eavclk tdh ncrst twe hquitcels en tdhsh tdrhh hquitcels, wh kht x 6 2

9, y 6

2

9.

Zdhsh viauhs iase sitcsny tdh tdcrj hquitcel. Dhloh tdh kcvhl neur peclts irh oepailir.


(?;) Cn j,o,b,i

irh ily neur vhoters cl ?-jcfhlscelia spioh wctd tdh sifh clctcia peclt ilj suod


21/26

FIZD_


tdit 0j2ob2i?

, sdew tdit tdh thrfclia I, B, O, J en tdhsh vhoters irh oepailir. Ncljtdh peclt (P) it wdcod IO ilj BJ fhht. Iase nclj tdh ritce cl wdcod P jcvcjhs IO ilj BJ.

(?4) _dew tdit tdh vhoter obi

, iob

ilj o:b?i2

irh lel-oepailir, wdhrh o,b,i

irh

ily lel-oepailir vhoters.

(?3) Nclj tdhviauh en ner wdcod tdh neur peclts wctd pesctcel vhoters gm , gm8c: , g:m3c?

ilj g:m:c: irh oepailir..

Ilswhrs = (?;):

oi?p

P jcvcjhs IO cl 9 = ? ilj BJ cl 9 = 9 ritce (?3) 6 9

Ippacoitcel en vhoters =(i) [erg jelh ikiclst i oelstilt neroh

N evhr i jcspaiohfhlt

s cs jhnclhj is s.N[

(b) Zdh tilkhltcia vhaeocty

^ en i bejy fevclk cl i ocroah cs kcvhl by rx^

, wdhrh

r cs tdh pesctcelvhoter en tdh peclt P.

(o) Zdh fefhlt enN ibeut E cs jhnclhj is rwdhrh,NxrF

cs tdh pesctcel vhoter en P w.r.t. E. Zdh

jcrhotcel en

F cs iaelk tdh lerfia te tdh pailh EPL suod tdit r N F, & nerf i rckdt diljhj systhf.

(j) Fefhlt en tdh oeupah 6 ( ) r r x N9 2 , wdhrh 29 riljr

irh pesctcel vhoters en tdh peclt en tdh ippacoitcel

en tdh nerohs .NiljN

Hxifpah# ?9= Nerohs en fiklctujhs 8 ilj ? ulcts iotclk cl tdh jcrhotcels g?m2c> ilj g>m2c?

rhsphotcvhay iot el i pirtcoah wdcod cs jcspaiohj nref tdh peclt (2, 2, 9) te (:, ?, 9). Nclj tdhwerg jelh by tdh nerohs.

_eautcel = Aht N

bh tdh rhsuatilt neroh ilj j

bh tdh jcspaiohfhlt vhoter. Zdhl,

N

63:?>

)g?m2c>(8

+ ?

?>:3

)g>m2c?(

6

;

9)g??m:c?3(

ilj j

6 )gm?c:( )gm2c2( 6 g2mc2

Zetia werg jelh 6 N

. j

6;

9)g??m:c?3( . )g2mc2(

6 ;

9(;4 + : + >>) 6 ;

9:4ulcts.



22/26


23/26

FIZD_

"filcsdgufirpdyscos.cl" 2?

Hxifpah#?? = C n o,b,i

irh tdrhh futuiaay phrphljcouair vhoters en hquia fiklctujh, prevh tdit obi

cs

hquiaay cloaclhj wctd vhoters b,i

ilj o

.

_eautcel = Aht |i|

6 |b|

6 |o|

6 (siy). _cloh o,b,i

irh futuiaay

phrphljcouair vhoters, tdhrhnerh b.i

6 o.b

6 i.o

6 0 ..............(c)

Lew,

2

obi

6 i.i

+ b.b

+ o.o

+ b.i2

+ o.b2

+ i.o2

6 2|i|

| + 2|b|

+ 2|o|

WXsclk (c) V

6 ?2 W |i|

6 |b|

6 |o|

6V

|obi|

6 ? ..............(cc)

_uppesh obi

fighs ilkahs9,

2,

? wctd b,i

ilj o

rhsphotcvhay. Zdhl,

oes9

6|obi||i|

)obi(.i

6

|obi||i|

o.ib.ii.i

6|obi||i|

|i| 2

6

|obi|

|i|

6 ?

6?

9WXsclk (cc)V

9

6 oes9

?

9

_cfcairay,2

6 oes9

?

9ilj

?6 oes9

?

9

9

6 2

6 ?.

Dhloh, obi

cs hquiaay cloaclhjhj wctd b,i

ilj o

Hxifpah#?: = Prevh usclk vhoters = Cn twe fhjcils en i trcilkah irh hquia, tdhl ct cs csesohahs.

_eautcel = Aht IBO bh i trcilkah ilj aht BH ilj ON bh twe hquia fhjcils. Zigclk I is tdh erckcl, aht tdh

pesctcel vhoters en B ilj O bh b

ilj o

rhsphotcvhay. Zdhl,

P.^. en H 62

9o

ilj P.^. en N 62

9b

BH 6 29

)b2o(

ON 6 2

9)o2b(

Lew, BH 6 ON |BH| 6 |ON|

2|BH| 6 2|ON| 2

)b2o(2

9 6

2

)o2b(2

9

:

92|b2o|

6

:

92|o2b|

2|b2o|

6 2|o2b|

)b2o(

. )b2o(

6 )o2b(

. )o2b(

o.o o.b:

+ b.b:

6 b.b

o.b:

+ o.o:

2|o|

o.b:

+ : 2|b|

6 2|b|

o.b:

+ 2|o|:


24/26

FIZD_

"filcsdgufirpdyscos.cl" 2:

? 2|b|

6 ? 2|o|

2|b|

6 2|o|

IB 6 IO Dhloh trcilkah IBO cs il csesohahs trcilkah.

Hxifpah#?8 = Xsclk vhoters = Prevh tdit oes (I + B) 6 oes I oes B scl I scl B

_eautcel = Aht ET ilj ES bh tdh oeerjclith ixhs ilj aht c ilj m bh ulct vhoters iaelk ET ilj ES

rhsphotcvhay. AhtTEP 6 I iljTEY 6 B. Jriwl PAET ilj YF ET.

Oahiray ilkah bhtwhhl EP ilj EY cs I + B

ClEAP, EA 6 EP oes I ilj AP 6 EP scl I. Zdhrhnerh EA 6 (EP oes I) c ilj

AP 6 (EP scl I) m

Lew, EA + AP 6 EP

EP 6 EP W(oes I ) c (scl I)I) m V ......(c)ClEFY, EF 6 EY oes B ilj FY 6 EY scl B.

Zdhrhnerh, EF 6 (EY oes B) c , FY 6 (EY scl B) m

Lew, EF + FY 6 EY

EY 6 Vm)B(sclc)BW(oesEY ......(cc)Nref (c) ilj (cc), wh kht

EP . EY 6 EP W(oes I) c (scl I)I) m V . EY W(oes B) c + (scl B) m V

6 EP . EY Woes I oes B scl I scl BV

But, EP . EY 6 |EP| |EY| oes (I + B) 6 EP . EY oes (I + B)

EP . EY oes (I + B) 6 EP . EY Woes I oes B scl I scl BV oes (I + B) 6 oes I oes B scl I scl B

Hxifpah#?> = Prevh tdit cl ily trcilkah IBO

(c) o2 6 i2 + b2 2ib oes O (cc) o 6 boes I + ioes B.

_eautcel = (c) ClIBO, IB + BO + OI 6 0

BO + OI 6 IB ......(c)_quirclk betd scjhs

( BO )2 + ( OI) 2 + 2 ( BO ). OI 6 (IB)2

i2 + b2 + 2 ( BO . OI) 6 o2 o2 6 i2 + b2 + 2 ib oes ( O) o2 6 i2 + b2 2ib oesO

(cc) ( BO + OI). IB 6 IB. IB

BO . IB + OI. IB6 o2

io oesB bo oes I 6 o2

ioesB + boesI 6 o.

Hxifpah#?; = Cn J, H, N irh tdh fcj-peclts en tdh scjhs en i trcilkah IBO, prevh by vhoter fhtdej tdit irhi en

JHN 6:

9(irhi enIBO)

_eautcel = Zigclk I is tdh erckcl, aht tdh pesctcel vhoters en B ilj O bh b

ilj o

rhsphotcvhay. Z dhl tdh

pesctcel vhoters en J, H ilj N irh29 )ob( ,

29 o ilj

29

b rhsphotcvhay..


25/26

FIZD_


Lew, JH 62

9o

2

9)ob(

6

2

b

ilj JN 62

9b

2

9( )ob(

6

2

o

^hoter irhi enJHN 6 6

6 6 6 (vhoter irhi en IBO)

Dhloh irhi enJHN 6 irhi en IBO.

Hxifpah#?4= P, Y irh tdh fcj-peclts en tdh lel-piriaaha scjhs BO ilj IJ en i triphzcuf IBOJ. _dew tdit

IPJ 6OYB.

_eautcel = Aht 6 ilj 6

Lew JO cs piriaaha te IB tdhrh hxcsts i so iair t suod tdit 6 t 6 t

6 + 6

Zdh pesctcel vhoters en P ilj Y irh ilj rhsphotcvhay..

Lew 2 6

6 6 (9 + t)

Ias e 2 6 6

6 6

6 6 6

Dhloh Prevh.

Hxifpah#?3 = Aht ilj irh ulct vhoters ilj cs i vhoter suod tdit 6 ilj tdhl nclj

tdh viauh en ._eautcel = Kcvhl 6 ilj

6 (is )

(usclk . 6 9 ilj 6 0, scloh ulct vhoter)

6 0 (is 0) . ... .. .. .. .. .( c)

Lew

(kcvhl + u)

(is nref (c))


26/26

FIZD_

.

0 (is 6 0)

9 (is 6 6 9)

6 9

Hxifpah#:0 = Cl ily trcilkah, sdew tdit tdh phrphljcouair bcshoters en tdh scjhs irh oelourrhlt.

_eautcel = Aht IBO bh tdh trcilkah ilj J, H ilj N irh rhsphotcvhay fcjjah peclts en scjhs BO, OI ilj

IB. Aht tdh phrphljcouair bcshoters en BO ilj OI fhht it E. Mecl EN. [ h irh rhqucrhj te

prevh tdit EN cs te IB. Aht tdh pesctcel vhoters en I, B, O wctd E is erckcl en rhnhrhloh

bh , ilj rhsphotcvhay..

6 ( + ), 6 ( + ) ilj 6 ( + )

Iase 6 , 6 ilj 6

_cloh EJ BO

( + ) . ( ) 6 0

b2 6 o2 ............(c)_cfcairay EH OI

( + ) . ( ) 6 0

i2 6 o2 ............(cc)nref (c) ilj (cc) wh divh i2 b2 6 0

( + ) . ( ) 6 0 ( + ) . ( ) 6 0

Hxifpah#:9= I, B, O, J irh neur peclts cl spioh. usclk vhoter fhtdejs, prevh tdit

IO2 + BJ2 + IJ2 + BO2 IB 2 + OJ2 wdit cs tdh cfpacoitcel en tdh sckl en hquiacty.

_eautcel = Aht tdh pesctcel vhoter en I, B, O, J bh ilj rhsphotcvhay tdhl

IO2 + BJ2 + IJ2 + BO2 6 . + . + . + .

6 + 2 + + 2 + + 2 + +

6 + 2 + + 2 + + + +

+ 2 +

6 . + . +

6 IB2 + OJ2 + .

IB 2 + OJ2

IO2 + BJ2 + IJ2 + BO2 IB2 + OJ2

ner tdh sckl en hquiacty te deaj, 6 0

ilj irh oeaaclhir, tdh neur peclts I, B, O, J irh oeaaclhir

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Vector Theory E